Equation of the Month

Equation of the Month

A blog run by the

Theoretical Population Ecology and Evolution Group,

Biology Dept.,

Lund University

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

Tuesday, June 28, 2011

The Canonical Equation of Adaptive Dynamics

What it means
The equation describes how the value of an ecological trait (z) evolves depending on the per capita mutation rate (μ), the variance of mutation effects (σ2), the population size at equilibrium (N* ) and the selection gradient (the last factor). W(z', z) is called invasion fitness and is measured as the per capita growth rate of a morph with trait value z' in an environment where a morph with a trait value z dominates (the resident trait). Note that the derivative in the last factor, i.e. the slope of the invasion fitness, is taken with respect to the mutant trait z' and evaluated at the resident trait value z.

Implications and importance
The equation is derived for mutation-limited evolution in large, monomorphic, asexual populations (Dieckmann and Law 1996). Changes in the trait value are assumed to be small and occur as successful mutant populations establish and replace the resident population. It is biologically straightforward to see why the different factors in the equation affect the rate of evolution. To start with, the product of and N* dictates how often mutations arise in the population. The factor 1/2 appears in the equation because under directional selection half of the mutations in a one-dimensional trait are bound to go in the 'wrong' direction. Higher variance in the mutation effects (σ;2) increases the rate of evolution by making the mutational steps longer. The slope of the invasion fitness around the resident trait value indicates how much fitness increases (or decreases) with a small mutational step. Evolution will be faster with a steeper slope since the likelihood of a successful invasion increases when the relative fitness advantage of the mutant over the resident is high. This slope also affects the direction of evolution such that z evolves towards higher values when the slope is positive and vice versa. Evolution will come to a halt when the slope of the invasion fitness is zero.

The canonical equation of adaptive dynamics, and generalisations of it, is especially useful for dealing with frequency-dependent selection and situations where the ecological feedback environment is affected by the evolutionary change. It has strong connections to evolutionary game theory and can for example be used to study gradual evolution to an Evolutionary Stable Strategy, ESS, or to evolutionary branching points (Geritz et al 1998, McGill and Brown 2007). Applications include food web evolution, speciation, fisheries management and the evolution of cooperation.

The canonical equation is related to other approaches to describe gradual evolution such as quantitative genetics or strategy dynamics. The approaches differ mainly in assumptions about the genetic variation (e.g. mutation-limited evolution vs. standing genetic variation) and whether or not ecological feedback is affecting evolution or not (changing vs. fixed adaptive landscape).

Jacob Johansson

Further reading:
Dieckmann U. and Law R. 1996. The dynamical theory of coevolution: A derivation from stochastic ecological processes. Journal of Mathematical Biology 34: 579–612

Geritz, S., É. Kisdi, E., G. Meszéna, G., and J. A. J. Metz, 1998. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolutionary Ecology 12: 35-57.

Champagnat, N., Ferrière, R., Ben Arous, G. (2001) The canonical equation of adaptive dynamics: a mathematical view. Selection 2, 73-83 .

Waxman, D. and Gavrilets, S. 2005. 20 Questions on Adaptive Dynamics. Journal of Evolutionary Biology 18: 1139-1154

McGill, B., and J. Brown. 2007. Evolutionary game theory and adaptive dynamics of continuous traits. Annual Review of Ecology, Evolution and Systematics 38: 403-435.

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